1. CJM 2013 (vol 66 pp. 505)
 Arapura, Donu

Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes
Suppose that $Y$ is a cyclic cover of projective space branched over
a hyperplane arrangement $D$, and that $U$ is the complement of the
ramification locus in $Y$. The first theorem implies that the
BeilinsonHodge conjecture holds for $U$ if certain multiplicities of
$D$ are coprime to the degree of the cover. For instance this applies
when $D$ is reduced with normal crossings. The second theorem shows
that when $D$ has normal crossings and the degree of the cover is a
prime number, the generalized Hodge conjecture holds for any toroidal
resolution of $Y$. The last section contains some partial extensions
to more general nonabelian covers.
Keywords:Hodge cycles, hyperplane arrangements Category:14C30 

2. CJM 2013 (vol 66 pp. 1225)
3. CJM 2011 (vol 63 pp. 1038)
 Cohen, D.; Denham, G.; Falk, M.; Varchenko, A.

Critical Points and Resonance of Hyperplane Arrangements
If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement
$\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship
between the critical set of $\Phi_\lambda$, the variety defined by the vanishing
of the oneform $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$.
For arrangements satisfying certain conditions, we show that if $\lambda$ is
resonant in dimension $p$, then the critical set
of $\Phi_\lambda$ has codimension
at most $p$. These include all free arrangements and all rank $3$ arrangements.
Keywords:hyperplane arrangement, master function, resonant weights, critical set Categories:32S22, 55N25, 52C35 

4. CJM 2005 (vol 57 pp. 416)
 Wise, Daniel T.

Approximating Flats by Periodic Flats in \\CAT(0) Square Complexes
We investigate the problem of whether every immersed flat plane in a
nonpositively curved square complex is the limit of periodic flat
planes. Using a branched cover, we reduce the problem to the case of
$\V$complexes. We solve the problem for malnormal and cyclonormal
$\V$complexes. We also solve the problem for complete square
complexes using a different approach. We give an application towards
deciding whether the elements of fundamental groups of the spaces we
study have commuting powers. We note a connection between the flat
approximation problem and subgroup separability.
Keywords:CAT(0), periodic flat planes Categories:20F67, 20F06 

5. CJM 2000 (vol 52 pp. 123)
 Harbourne, Brian

An Algorithm for Fat Points on $\mathbf{P}^2
Let $F$ be a divisor on the blowup $X$ of $\pr^2$ at $r$ general
points $p_1, \dots, p_r$ and let $L$ be the total transform of a
line on $\pr^2$. An approach is presented for reducing the
computation of the dimension of the cokernel of the natural map
$\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl(
\CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$
to the case that $F$ is ample. As an application, a formula for
the dimension of the cokernel of $\mu_F$ is obtained when $r = 7$,
completely solving the problem of determining the modules in
minimal free resolutions of fat point subschemes\break
$m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for
an arbitrary algebraically closed ground field~$k$.
Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group Categories:13P10, 14C99, 13D02, 13H15 
